Video report by ITV News Political Correspondent Romilly Weeks
Boris Johnson has written a letter to MPs saying he will tell the EU that "further delay is not a solution", despite losing a key Commons vote.
In his letter to all MPs and Peers, Mr Johnson wrote: "I have made clear that I do not want more delay.
"European leaders have made clear they do not want more delay. It is to my great regret that today the House has voted for more delay.
"The public want us to get Brexit done so the country can move on. The best thing for the United Kingdom and the European Union is for us to leave with this new deal on 31 October."
His comments come after a special Saturday sitting - the first in 37 years - where the Commons voted by 322 to 306, a majority of 16, in favour of an amendment intended to force the prime minister to seek a Brexit delay.
The Prime Minister added: "I will not negotiate a delay with the European Union. I will tell the EU what I have told the British public for my 88 days as Prime Minister: further delay is not a solution.
"That is why next week this Government will introduce the legislation needed for us to leave the European Union with our great new deal on 31 October."
He continued: "It is quite possible that our friends in the European Union will reject Parliament's request for further delay (or not take a decision quickly).
"In these circumstances, I hope colleagues on all sides of the House will - faced with a choice of our new deal or no deal - support this new deal."
Commons Leader Jacob Rees-Mogg said earlier that the Government is planning to give MPs a chance to have a meaningful vote on the Prime Minister's Brexit deal on Monday.
After the result of the vote, the House of Commons Twitter account posted that the government now "must ask for an extension of Article 50 under the Benn Act and set out how it intends to proceed".
Under the terms of the so-called Benn Act, if he does not have agreement on a deal on Saturday, the Prime Minister is required to seek a further extension to the Article 50 withdrawal process until the end of January.
But a Number 10 source said Mr Johnson will not ask for a delay to Brexit.
A source revealed the government's plan to avoid an extension would be to characterise the request for a three-month Brexit delay as coming from Parliament, not from the prime minister.
The source said: "The prime minister will tell EU leaders there should be no delays, they should reject Parliament's letter asking for a delay and we should get Brexit done on October 31."
Speaker of the House of Commons John Bercow said he would would sign a letter asking for a Brexit delay if Mr Johnson failed to do so himself and if the courts or Parliament asked him to.
Sir Oliver Letwin, who tabled the amendment, claimed its purpose was to prevent an unintended no-deal exit happening by default if the deal had not been fully ratified by the October 31 deadline.
Mr Letwin, a former Cabinet minister who had the Tory whip withdrawn after rebelling over Brexit, said the amendment was an "insurance policy" to prevent Britain "crashing out" without a deal on October 31.
The division list for the Letwin amendment showed 231 Labour MPs voted for it alongside the 10 DUP MPs.
They were joined by 19 Liberal Democrats, 35 SNP MPs, 17 Independents, four Plaid Cymru MPs, five Independent Group for Change MPs and Green Party MP Caroline Lucas.
The list added 283 Conservative MPs opposed it along with six Labour MPs and 17 Independents.
As ITV News Political Editor Robert Peston says, the PM is "paying a heavy price for alienating his Northern Irish partners in government", because their 10 votes could have swayed it in his favour.
The DUP's Arlene Foster and Nigel Dodds said they could not support the prime minister's deal due to "what is being suggested on customs and consent issues" and that there is a "lack of clarity on VAT".
Ministers have signalled that they will press ahead with plans to table legislation relating to the Brexit deal next week with a view to securing Britain's departure by the end of the month.
MPs backing the amendment means "Super Saturday" - as it had been dubbed - is no longer the definitive moment the PM wanted it to be.
Labour Leader Jeremy Corbyn said he welcomed MPs voting for the Letwin amendment.
He added: "I welcome today's vote, it's an emphatic decision by this House that has declined to back the Prime Minister's deal today and clearly voted to stop a no-deal crash out from the European Union.
"The Prime Minister must now comply with the law. He can no longer use the threat of a no-deal crash out to blackmail members to support his sell-out deal."
The SNP leader at Westminster Ian Blackford said that if Mr Johnson acted as if we was "above the law", he would find himself in court.
Liberal Democrat leader Jo Swinson said: "The most urgent thing right now is the Prime Minister complies with the law."
The European Commissions' chief spokeswoman Mina Andreeva urged the Government "to inform us about the next steps as soon as possible".
In full: MPs debate Boris Johnson's Brexit deal: